Inference of Several Log-normal Distributions

Transcription

1 Inference of Several Log-normal Distributions Guoyi Zhang 1 and Bose Falk 2 Abstract This research considers several log-normal distributions when variances are heteroscedastic and group sizes are unequal. We proposed fiducial generalized pivotal quantities (FGPQ)-based simultaneous confidence intervals for ratios of the means. We also proved that the constructed confidence intervals have correct asymptotic coverage. Simulation results show that the proposed methods work well. This research together with overall tests by Li (2009) provide a solution of inference on several log-normal distributions. Key Words: Log-normal, Fiducial Generalized Pivotal Quantities (FGPQ), Multiple Comparison, Simulations, Unequal Variances. 1 Introduction Log-normal distribution is widely used to describe the distribution of positive random variables that exhibit skewness in biological, medical, and economical studies. The problem of equality and multiple comparisons of the group means are common interests in many observational and experimental data arising from several populations. Unfortunately, if sample variances are unequal, the standard ANOVA tests don t apply for log-normal distributions even after transformation, since the null hypothesis based on log-transformed outcomes is not equivalent to the one based on the original outcomes (Zhou, Gao, & Hui, 1997). 1 Guoyi Zhang: Corresponding author, Assistant Professor, Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM, (gzhang123@gmail.com) 2 Bose Falk, Graduate student, Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM,

2 The problem of comparing means of several log-normal populations has been well studied in literature. Approximation procedures are commonly used regarding this problem, for example, Alexander-Govern test (1994), the Welch test (1951) and the James second-order test (1951) etc. These three approximate tests behave similarly, and perform better than the ANOVA F-test (Guo & Luh, 2000). Later, Gupta and Li (2006) presented a score test. Weerahandi (1993), Krishnamoorthy and Mathew (2003) investigated inferences on the means of log-normal distributions by generalized p-values and generalized confidence intervals. Li (2009) proposed a new generalized p-value procedure. Simulation results show that Li (2009) s methods perform better than other existing methods with respect to sizes and powers under different settings. Simultaneous confidence intervals for certain log-normal parameters are useful in pharmaceutical statistics. It is often of interest to compare the mean responses of two drugs to ensure that they are (more or less) equally effective. For example, twenty-three healthy male subjects each followed randomly allocated sequences of five treatments (with one week washout period between different treatments to ensure no carry-over effects), either no treatment or one of four active treatments from the same drug class used to treat the same illness (Bradstreet & Liss, 1995). One of the subjects are missing under treatment 2, which leads to an unbalanced case. Since data followed a log-normal distribution, to find out if there are any of the four similar to no treatment, or similar to each other, we require a new method on multiple comparisons for several log-normal distributions. Hannig, Iyer, and Patterson (2006) introduced a subclass of Weerahandi s generalized pivotal quantities, called FGPQs, and provided procedures to derive FGPQs. Hannig, E, Abdel-Karim, and Iyer (2006) proposed simultaneous fiducial generalized confidence intervals for ratios of means of log-normal distributions. Xiong and Mu (2009) proposed two kinds of simultaneous intervals based on FGPQ for all pairwise comparisons of treatment means in a one-way layout under heteroscedasticity. Xiong and Mu (2009) pointed out that if sample sizes are sufficiently large, Hannig, E, et al. (2006) s simultaneous confidence intervals are equal to one of their proposed intervals. Otherwise, Xiong and Mu (2009) methods perform better than Hannig, 2

3 E, et al. (2006) s methods. Following Xiong and Mu (2009) s idea, we proposed FGPQ-based simultaneous confidence intervals for all-pairwise comparisons for ratios of means from several log-normal populations under heteroscedasticity. Research in this paper together with overall tests proposed by Li (2009) provide a solution of several log-normal distributions. This paper is organized as follows. In Section 2, we review notation of generalized variable approach and overall tests by Li (2009). In Section 3, we propose FGPQ-based simultaneous confidence intervals for ratios of means from several log-normal distributions. In Section 4, we present simulation studies. Section 5 gives conclusions. 2 Background 2.1 Generalized variable approach Let Y ij, i = 1,, k, j = 1,, n i be a random sample from k log-normal distributions with parameters µ i and σ 2 i, and let X ij = logy ij. By definition, X ij is an independent random sample from the k populations and has a normal distribution of N(µ i, σ 2 i ). For each sample, the sample mean and variance are defined as follows ni j=1 X i = X ij, Si 2 = 1 n i (X ij n i n i 1 X i ) 2. j=1 Let Z i = n i ( X i µ i )/σ i and U 2 i = (n i 1)S 2 i /σ 2 i. It is well known that Z i N(0, 1) and U 2 i each population, define χ 2 (n i 1) and they are jointly independent. For M i = E(Y ij ) = e µ i+σ 2 i /2 and θ i = log(m i ) = µ i + σ 2 i /2. (1) Krishnamoorthy and Mathew (2003) suggested the following generalized pivotal variables for µ i and σi 2 : T µi = x i X i µ i S i / s i / Z i n i = x i n i U i / n i 1 s i/ ni 1 n i = x i Zis i, (2) n i U i 3

4 and T σ 2 i = s2 i σ 2 Si 2 i = where x i and s 2 i are the observed values of Xi and S 2 i. s 2 i U 2 i /(n i 1), (3) 2.2 Overall testing Li (2009) proposed overall tests for k log-normal populations. Simulation results show that Li (2009) s methods perform better than other existing methods with respect to sizes and powers under different scenarios. In this section, we briefly review Li s overall tests. By definition (1), the overall test of equality of group means H 0 : M 1 = M 2 = = M k vs H α : M is are not all equal, is equivalent to the hypothesis test H 0 : θ 1 = θ 2 = = θ k vs H α : θ is are not all equal. (4) Let H be a (k 1) k matrix with H =, θ = (θ 1, θ 2,, θ k ), X = ( X 1, X 2,, X k ), S 2 = (S1, 2 S2, 2, Sk 2), x = ( x 1, x 2,, x k ), and s 2 = (s 2 1, s 2 2,, s 2 k ). Using (2) and (3), the generalized pivotal quantities for θ i and Hθ are as follows, T θi = T µi + T σ 2 i /2 and T Hθ = H(T θ1,, T θk ). (5) The conditional mean and variance of pivotal quantity T Hθ can be found by µ T = E(T Hθ ( x, s 2 )) = HE(T θ ( x, s 2 )) 4

5 and Σ T = cov(t Hθ ( x, s 2 )) = Hcov(T θ ( x, s 2 ))H. The generalized p-value for testing (4) is given by p = P { (T Hθ µ T ) Σ 1 T (T Hθ µ T ) µ T Σ 1 T µ } T. (6) The following algorithm is given to calculate the generalized p-value i. Algorithm 1: For a given sample with group sizes (n 1,, n k ), group means ( x 1,, x k ), and variances (s 2 1,, s 2 k ), For l = 1,, L: Generate Z i and U 2 i, i = 1,, k Compute T l = T Hθ = H(T θ1,, T θk ) using (5). Compute ˆµ T = 1 L T l and L ˆΣ T = 1 L (T l ˆµ T )(T l ˆµ T ). L 1 l=1 Compute ˆ T 2 l = (T l ˆµ T ) ˆΣ 1 T (T l ˆµ T ), l = 1,, L and ˆ µ 0 2 = ˆµ T Let W 1 = 1 if ˆ T 2 l ˆ µ 0 2, else W 1 = 0. (end loop) 1 L W l is a simulated estimate of generalized p-value for testing (4). L l=1 3 Simultaneous pairwise comparisons l=1 ˆΣ 1 T ˆµ T. In this section, we propose FGPQ-based simultaneous confidence intervals for all-pairwise comparisons of means from the k log-normal populations under heteroscedasticity. testing problem is as follows H 0 : M i = M j for all i j vs H α : at least one of M i M j. (7) The Define ratio of the mean as M ij = M i /M j and θ ij = logm ij = log M i = log eµi+σ2 M j e = (µ µ j+σj 2/2 i + σ2 i 2 ) (µ j + σ2 j 2 ). i /2 5

6 The problem of constructing simultaneous confidence intervals for M ij is equivalent to the problem of constructing simultaneous confidence intervals for θ ij. The multiple comparison problem in (7) is equivalent to the hypothesis tests H 0 : θ ij = 0 vs H α : not all θ ij = 0. (8) Follow Xiong and Mu (2009), we define the FGPQs for µ i and σ 2 i for i = 1,, k as follows R µi = X i ni 1 The pivotal variable for θ i follows immediately as n i SiZ i, R U σ 2 i = (n i 1)Si 2, i = 1,, k. i Ui 2 R θi = R µi + R σ 2 i 2 = X i ni 1 n i SiZ i U i + (n i 1)S 2 i 2U 2 i. Consequently, R θij = R θi R θj = X i X j ni 1 n i SiZ i U i + n j 1 SjZ j n j U j + (n i 1)S 2 i 2U 2 i (n j 1)S 2 j 2U 2 j The conditional expectation and variance of R θij can be derived as follows η ij = E(R θij X, S 2 ) = X i X j + n i 1 2(n i 3) S2 i n j 1 2(n j 3) S2 j V ij = Var(R θij X, S 2 ) = + n i 1 (n i 1) 2 n i (n i 3) S2 i + 2(n i 3) 2 (n i 5) S4 i n j 1 (n j 1) 2 n j (n j 3) S2 j + 2(n j 3) 2 (n j 5) S4 j Let ξ ij be the variance of η ij, and let R ξij be the pivotal variable of ξ ij. We have the following equations: ξ ij = Var { E(R θij X, S 2 ) } ( ) 2 ( ) = σ2 i + σ2 j ni 1 2σ 4 2 i nj 1 2σj n i n j 2(n i 3) n i 2(n j 3) n j = σ2 i n i + (n i 1) 2 2n i (n i 3) 2 σ4 i + σ2 j n j + (n j 1) 2 2n j (n j 3) 2 σ4 j. 6

7 and R ξij = (n i 1)S 2 i n i U 2 i + (n j 1)S 2 j n j U 2 j + (n ( i 1) 2 (ni 1)Si 2 2n i (n i 3) 2 Ui 2 + (n ( j 1) 2 (nj 1)Sj 2 2n j (n j 3) 2 U 2 j ) 2 ) 2 FGPQs can be used to provide effective approximations to distributions (Xiong & Mu, 2009). The distribution of max i<j θ ij E(R θij X, S 2 ) Var(R θij X, S 2 ) can be approximated by the conditional distributions of Q = max i j R θij E(R θij X, S 2 ) Rξij. (9) Let q(α) be the conditional upper αth quantile of the distribution of Q. The (1 α)100% simultaneous confidence intervals for θ ij are η ij ± q(α) V ij for all i < j. (10) The following Theorem shows that the confidence intervals (10) have asymptotically correct coverage probabilities. Proof is similar as Xiong and Mu (2009) and is included in Appendix. Theorem 1. Let X i1,, X ini, i = 1,, k be random samples from k different populations and be mutually independent. Assume that 0 < σ 2 i = V ar(x i1 ) <, µ i = E(X i1 ), N = k i=1 n i and n i N λ i (0, 1) as N for all i, then P (θ ij η ij ± q(α) V ij for all i < j) p 1 α. We propose Algorithm 2 for constructing the simultaneous confidence intervals. Algorithm 2: For given observations y ij, i = 1,, k, j = 1,, n i, compute x ij = lny ij, i = 1,, k, j = 1,, n i 7

8 Compute x i and s 2 i, i = 1,, k For l = 1, 2,, L Generate Z i and Ui 2, i = 1,, k Compute R θij, R ξij, and Q l. End l loop. Compute q(α), the (1 α)100% percentile of Q l. 4 Simulations In this section, we use simulations to study the overall test and multiple comparisons of k log-normal distributions under the assumption of heteroscedastic variances and unequal sizes. The simulation settings follow from Li (2009). Statistical software R is used for all computations. The sample statistics x i and s 2 i are generated independently as x i N(0, σ 2 i /n i ) and s 2 i σ 2 i χ 2 n i 1/(n i 1), with 0 < σ 2 i 1, i = 2,, k. The simulation study was performed with factors: (1) number of levels k: k = 3 and k = 6; (2) population variance σ = (σ 2 1,, σ 2 k ): various combinations; (3) population mean µ = (µ 1,, µ k ): various combinations; (4) Significance level α: 0.01, 0.05 and 0.1; (5) group sizes n = (n 1,, n k ): various combinations. For a given sample size and parameter configuration, we generated 2000 observed vectors ( x 1,, x k, s 2 1,, s 2 k ) and used 5000 runs to estimate the Type 1 errors (simulated p-value). According to our experience, 5000 runs is sufficient to guarantee the precision of simulated p-value. We consider both overall tests (Li, 2009) and multiple comparisons (proposed method). Algorithm 1 is used to estimate the p-value of overall test (4). Algorithm 2 is used to find q α, the 1 α percentile of the simulated distribution of Q l. Using a desktop with Intel core 2Dup CPU 2.66 GHz with 4GB memory, for each generated data, when k = 3, Algorithm 1 takes no more than 1 second and Algorithm 2 takes no more than 2 seconds; when k = 6, Algorithm 1 takes no more than 1 second and Algorithm 2 takes no more than 6 seconds. 8

9 In Tables 1, 2, 3, and 4, the following notation applies. n = (n 1,, n k ), k = 3 or 6 is a vector of unequal group sizes. For k = 3, we have 1 = (10, 16, 20), 2 = (10, 10, 10), 3 = (20, 16, 10), 1 = (50, 80, 100), n(3) 2 = (50, 50, 50), and n(3) 3 = (100, 80, 50). For k = 6, we have 1 = (10, 12, 12, 16, 16, 20), 2 = (10, 10, 10, 10, 10, 10), 3 = (20, 16, 16, 12, 12, 10), 1 = (50, 60, 60, 80, 80, 100), 2 = (50, 50, 50, 50, 50, 50), and n(6) 3 = (100, 80, 80, 60, 60, 50). Note that n (j) i /n(j) i = 5, i = 1, 2, 3, j = 3 or 6. µ = (µ 1,, µ k ) k = 3 or 6 is a vector of unequal means. For k = 3, we consider µ (3) 1 = (1, 1, 1), µ (3) 2 = (1, 1, 1.25), µ (3) 3 = (1, 1.25, 1.45). For k = 6, we consider µ (6) 1 = (1, 1, 1, 1, 1, 1), µ (6) 2 = (1, 1, 1, 1, 1, 0.8), µ (6) 3 = (1, 1, 1, 1, 1, 0.9). σ = (σ1, 2, σk 2 ), k = 3 or 6 is a vector of unequal variances with σ(3) 1 = (0.1, 0.1, 0.1), σ (3) 2 = (1, 1, 0.5), σ (3) 3 = (1, 0.5, 0.1), σ (6) 1 = (0.1, 0.1, 0.1, 0.1, 0.1, 0.1), σ (6) 2 = (0.1, 0.1, 0.1, 0.1, 0.1, 0.5), and σ (6) 3 = (0.1, 0.1, 0.1, 0.1, 0.1, 0.3). To simplify notation in the tables, let C (3) 1 = (µ (3) 1, σ (3) 1 ), C (3) 2 = (µ (3) 2, σ (3) 2 ), C (3) 3 = (µ (3) 3, σ (3) 3 ), C (6) 1 = (µ (6) 1, σ (6) 1 ), C (6) 2 = (µ (6) 2, σ (6) 2 ), and C (6) 3 = (µ (6) 3, σ (6) 3 ). Tables 1, 2, 3 and 4 report the simulation results of Li (2009) s overall test and proposed multiple comparison procedure under different settings. We can see that simulated p-values of overall test and multiple comparison procedure are close to the nominal levels when the group sizes are 10 or more. When group sizes increased by five times, i.e., n (j) i /n(j) i = 5, the simulated p-value comes slightly closer to the nominal level in general, but no significant difference observed. Notice that 2, 2, n(6) 2, and 2 are with equal group sizes, and C(3) 1 and C (6) 1 are with equal variances. We found that the overall tests and the proposed MCP perform well for both unbalanced unequal variance and balanced equal variance cases. 5 Conclusions In this article, we proposed an FGPQ-based new method to construct simultaneous confidence intervals for ratios of means from several log-normal distributions under heteroscedasticity and unequal group sizes. Simulation studies show that these intervals perform well. We also proved that the constructed confidence intervals have correct asymptotic coverage. 9

10 Table 1: Simulation results of overall test from Li, 2009 for three groups. Numbers in Table are simulated p-values. α =.01 α =.05 α =.1 n C (3) 1 C (3) 2 C (3) 3 C (3) 1 C (3) 2 C (3) 3 C (3) 1 C (3) 2 C (3) Table 2: Simulation results of overall test from Li, 2009 for six groups. Numbers in Table are simulated p-values. α =.01 α =.05 α =.1 n C (6) 1 C (6) 2 C (6) 3 C (6) 1 C (6) 2 C (6) 3 C (6) 1 C (6) 2 C (6)

11 Table 3: Simulation results of the proposed FGPQ-based multiple comparison procedure for three groups. Numbers in Table are simulated p-values. α =.01 α =.05 α =.1 n C (3) 1 C (3) 2 C (3) 3 C (3) 1 C (3) 2 C (3) 3 C (3) 1 C (3) 2 C (3) Table 4: Simulation results of the proposed FGPQ-based multiple comparison procedure for six groups. Numbers in Table are simulated p-values. α =.01 α =.05 α =.1 n C (6) 1 C (6) 2 C (6) 3 C (6) 1 C (6) 2 C (6) 3 C (6) 1 C (6) 2 C (6)

12 The proposed methods could be applied to group mean comparisons when data are arising from log-normal distributions. Acknowledgements The authors thank the referee for helpful comments and constructive suggestions to improve the manuscript. 12

13 Appendix Proof of Theorem 1. Proof. By the central limit theorem, we have N ((η12 θ 12 ), (η 13 θ 13 ),, (η k 1,k θ k 1,k )) d N(0, U), where U is an k(k 1)/2 k(k 1)/2 positive definite matrix. Let u ab, a, b = 1, 2,, k(k 1)/2 be its (a, b)th entry. It can be shown that and almost surely. Therefore, ( η 12 θ 12 V12, η 13 θ 13 u aa = σ2 i λ i + σ4 i 2λ i + σ2 j λ j + σ4 j 2λ j NV ij σ2 i λ i + σ4 i 2λ i + σ2 j λ j + σ4 j 2λ j V13 ),, η k 1,k θ k 1,k d N(0, U ), Vk 1,k where the (a, b)th entry of U is u ab / u aa u bb. Take a random vector (Z 1, Z 2,, Z k(k 1)/2 ) distributed according to N(0, U ). By the continuous mapping theorem max i<j θ ij η ij Vij d max Z a for 1 a k(k 1)/2. For i = 1,, k, Ui 2 p /n i 1. For all i j, N(Rθij η ij) = { ni 1 N + (n i 1)S 2 i 2U 2 i conditionally on T = ( X, S 2 ) almost surely. n i SiZ i U i + (n j 1)S 2 j 2U 2 j n j 1 SjZ j n j U j n i 1 2(n i 3) S2 i + n j 1 2(n j 3) S2 j = σ i Z j σ i Z i + o p (1) (11) λi λi 13 }

14 Recall that NV ij σ2 i λ i + σ4 i 2λ i + σ2 j λ j + σ4 j 2λ j NR ξij = N (n i 1)S 2 i n i U 2 i + N (n j 1)S 2 i n j U 2 j almost surely and note that + N (n i 1) 2 2n i (n i 3) 2 + N (n j 1) 2 2n j (n j 3) 2 = σ2 i λ i + σ4 i 2λ i + σ2 j λ j + σ4 j λ j + o p (1) ( (ni 1)S 2 i U 2 i ( (nj 1)S 2 j conditionally on T almost surely. It can be shown that Equation (11) implies θ ij η ij d max i<j max Rξij 1 a k(k 1)/2 Z a (12) on T almost surely. Let F be the cumulative distribution function of max 1 a k(k 1)/2 Z a. By the continuity of F sup x F n (x T ) F (x) 0 almost surely, where F n is the conditional distribution function of the left side of (12). As a result, P ( θ ij η ij ± q(α) ) V ij for all i < j U 2 j ) 2 ) 2 ( ) } θ ij η ij = P {F n max i<j T 1 α Vij { ( ) } θ ij η ij = P F max i<j + o Rξij p (1) 1 α d 1 α References Alexander, R., & Govern, D. M. (1994). A new and simpler approximation and ANOVA under variance heterogeneity. Jounal of Educational Statistics, 19, Bradstreet, T. E., & Liss, C. L. (1995). Favorite data sets from early (and late) phases of drug research - part 4. Proceedings of the Section on Statistical Education of the American Statistical Association. 14

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